arXiv subjects

Richard D. Sandberg

Publications and source records attributed to Richard D. Sandberg.

14 recordsLinked to original sources

arXiv2026

Quantum-Inspired Computational Fluid Dynamics for Transient Turbulent Compressible Flows

Quantum-inspired algorithms are an emerging class of algorithms for computational fluid dynamics (CFD) with potentially favourable scaling for large problems compared to classical methods. However, their applications have been limited to incompressible flows due to arithmetic limitations, which are addressed in this work. This work introduces the first true end-to-end quantum-inspired computational fluid dynamics (QICFD) solver for direct numerical simulation of the compressible Navier-Stokes equations, that is, all arithmetic operations, as well as initialisation and post-processing, are undertaken in the tensor train (TT) format. Importantly, new division and square-root algorithms using TTs enable the use of Sutherland's law for viscosity, and leverages recent improvements in TT arithmetic algorithms for reduced runtime and memory cost. The new QICFD solver is validated by comparison with the classical CFD solver HiPSTAR and by way of a challenging fluid-flow test case, the low resolution Taylor-Green Vortex (TGV) at Mach numbers of 0.8 and 0.1. The TGV test case is a transient turbulent case that is very sensitive to accumulating errors, yet our QICFD solver achieves excellent agreement with the classical CFD reference. This work demonstrates the correctness of the new TT division and square-root algorithms, and that QICFD is capable of compressible flow simulations. The new QICFD solver is also able to perform simultaneous simulations, running multiple TGV-like cases initialised differently in parallel with reduced extra cost. Finally, the demonstrated TGV test case reveals additional challenges of QICFD as well as highlight the need for future advances to make TT methods viable for industrially-relevant conditions.

physics.flu-dyn↗

arXiv2026

Boundary layer transition induced by surface roughness distributed over a low-pressure turbine blade

Direct numerical simulations of a low-pressure turbine with roughness elements distributed over the blade surface have been performed. A series of fifteen cases with varying roughness heights and streamwise wavenumbers are introduced to present a systematic study of the effect of roughness on the various transition phenomena in the suction-side boundary layer. For cases with large roughness heights, the boundary layer is violently disturbed by the wake of rough elements in the leading edge (LE) region, and maintains the turbulent state over the whole blade suction-side. For cases with small roughness heights, however, the disturbances induced by the LE roughness are suppressed by the favourable pressure gradient in the downstream boundary layer, and the relaminarized flow does not undergo transition until the separation near the blade trailing edge (TE). Furthermore, the streamwise wavenumber of the distributed roughness plays an important role in cases with intermediate roughness height. Specifically, cases with larger streamwise slope show earlier transition induced by strong shear layer instability, which manages to suppress the mean flow separation near the TE region. Overall, the combined effect of several factors, including the geometric effect at the blade LE and TE, the complex pressure gradient distribution across the turbine vane, and the various roughness configurations, is responsible for the intriguing boundary layer behaviours in the present study.

physics.flu-dyn↗

arXiv2025

A Surrogate-Augmented Symbolic CFD-Driven Training Framework for Accelerating Multi-objective Physical Model Development

Computational Fluid Dynamics (CFD)-driven training combines machine learning (ML) with CFD solvers to develop physically consistent closure models with improved predictive accuracy. In the original framework, each ML-generated candidate model is embedded in a CFD solver and evaluated against reference data, requiring hundreds to thousands of high-fidelity simulations and resulting in prohibitive computational cost for complex flows. To overcome this limitation, we propose an extended framework that integrates surrogate modeling into symbolic CFD-driven training in real time to reduce training cost. The surrogate model learns to approximate the errors of ML-generated models based on previous CFD evaluations and is continuously refined during training. Newly generated models are first assessed using the surrogate, and only those predicted to yield small errors or high uncertainty are subsequently evaluated with full CFD simulations. Discrete expressions generated by symbolic regression are mapped into a continuous space using averaged input-symbol values as inputs to a probabilistic surrogate model. To support multi-objective model training, particularly when fixed weighting of competing quantities is challenging, the surrogate is extended to a multi-output formulation by generalizing the kernel to a matrix form, providing one mean and variance prediction per training objective. Selection metrics based on these probabilistic outputs are used to identify an optimal training setup. The proposed surrogate-augmented CFD-driven training framework is demonstrated across a range of statistically one- and two-dimensional flows, including both single- and multi-expression model optimization. In all cases, the framework substantially reduces training cost while maintaining predictive accuracy comparable to that of the original CFD-driven approach.

cs.LG↗

arXiv2025

Accelerating evolutionary exploration through language model-based transfer learning

Gene expression programming is an evolutionary optimization algorithm with the potential to generate interpretable and easily implementable equations for regression problems. Despite knowledge gained from previous optimizations being potentially available, the initial candidate solutions are typically generated randomly at the beginning and often only include features or terms based on preliminary user assumptions. This random initial guess, which lacks constraints on the search space, typically results in higher computational costs in the search for an optimal solution. Meanwhile, transfer learning, a technique to reuse parts of trained models, has been successfully applied to neural networks. However, no generalized strategy for its use exists for symbolic regression in the context of evolutionary algorithms. In this work, we propose an approach for integrating transfer learning with gene expression programming applied to symbolic regression. The constructed framework integrates Natural Language Processing techniques to discern correlations and recurring patterns from equations explored during previous optimizations. This integration facilitates the transfer of acquired knowledge from similar tasks to new ones. Through empirical evaluation of the extended framework across a range of univariate problems from an open database and from the field of computational fluid dynamics, our results affirm that initial solutions derived via a transfer learning mechanism enhance the algorithm's convergence rate towards improved solutions.

cs.NE↗

arXiv2023

Turbulence Model Development based on a Novel Method Combining Gene Expression Programming with an Artificial Neural Network

Data-driven methods are widely used to develop physical models, but there still exist limitations that affect their performance, generalizability and robustness. By combining gene expression programming (GEP) with artificial neural network (ANN), we propose a novel method for symbolic regression called the gene expression programming neural network (GEPNN). In this method, candidate expressions generated by evolutionary algorithms are transformed between the GEP and ANN structures during training iterations, and efficient and robust convergence to accurate models is achieved by combining the GEP's global searching and the ANN's gradient optimization capabilities. In addition, sparsity-enhancing strategies have been introduced to GEPNN to improve the interpretability of the trained models. The GEPNN method has been tested for finding different physical laws, showing improved convergence to models with precise coefficients. Furthermore, for large-eddy simulation of turbulence, the subgrid-scale stress model trained by GEPNN significantly improves the prediction of turbulence statistics and flow structures over traditional models, showing advantages compared to the existing GEP and ANN methods in both a priori and a posteriori tests.

physics.comp-ph↗

arXiv2022

Gradient Information and Regularization for Gene Expression Programming to Develop Data-Driven Physics Closure Models

Learning accurate numerical constants when developing algebraic models is a known challenge for evolutionary algorithms, such as Gene Expression Programming (GEP). This paper introduces the concept of adaptive symbols to the GEP framework by Weatheritt and Sandberg (2016) to develop advanced physics closure models. Adaptive symbols utilize gradient information to learn locally optimal numerical constants during model training, for which we investigate two types of nonlinear optimization algorithms. The second contribution of this work is implementing two regularization techniques to incentivize the development of implementable and interpretable closure models. We apply $L_2$ regularization to ensure small magnitude numerical constants and devise a novel complexity metric that supports the development of low complexity models via custom symbol complexities and multi-objective optimization. This extended framework is employed to four use cases, namely rediscovering Sutherland's viscosity law, developing laminar flame speed combustion models and training two types of fluid dynamics turbulence models. The model prediction accuracy and the convergence speed of training are improved significantly across all of the more and less complex use cases, respectively. The two regularization methods are essential for developing implementable closure models and we demonstrate that the developed turbulence models substantially improve simulations over state-of-the-art models.

physics.comp-ph↗

arXiv2021

Compressibility Effects on the Linear-stability of Centrifugal Buoyancy-induced Flow

The focus of this study is to understand the evolution of instability in centrifugal buoyancy-induced flow in a rotating system. The problem is of interest in atmospheric flows as well as in engineering applications. In this study, we perform direct numerical simulations (DNS) by solving the compressible Navier-Stokes equations and multi-dimensional stability analyses by using a forced DNS approach. We systematically and independently vary the Rayleigh and Mach numbers. The heat transfer by thermal conduction is used as base flow and maintained as a reference state, upon which the growth of small perturbations is investigated. It is found that the critical wavenumber obtained from the linear stability analysis at the onset of convection has a much shorter wavelength than the one that eventually appears in the non-linear regime. Further, the investigations show that compressibility effects lead to a reduction of the growth rate of the dominant mode, and it modifies the overall formation of convection cells in the cavity.

physics.flu-dyn↗

arXiv2021

Resolvent-based approach for H2-optimal estimation and control: an application to the cylinder flow

We consider estimation and control of the cylinder wake at low Reynolds numbers. A particular focus is on the development of efficient numerical algorithms to design optimal linear feedback controllers when there are many inputs (disturbances applied everywhere) and many outputs (perturbations measured everywhere). We propose a resolvent-based iterative algorithm to perform i) optimal estimation of the flow using a limited number of sensors; and ii) optimal control of the flow when the entire flow is known but only a limited number of actuators are available for control. The method uses resolvent analysis to take advantage of the low-rank characteristics of the cylinder wake and solutions are obtained without any model-order reduction. Optimal feedback controllers are also obtained by combining the solutions of the estimation and control problems. We show that the performance of the estimators and controllers converges to the true global optima, indicating that the important physical mechanisms for estimation and control are of low rank.

math.OC↗

arXiv2021

Data-driven algebraic models of the turbulent Prandtl number for buoyancy-affected flow near a vertical surface

The behaviour of the turbulent Prandtl number ($Pr_t$) for buoyancy-affected flows near a vertical surface is investigated as an extension study of {Gibson \& Leslie, \emph{Int. Comm. Heat Mass Transfer}, Vol. 11, pp. 73-84 (1984)}. By analysing the location of mean velocity maxima in a differentially heated vertical planar channel, we {identify an} {infinity anomaly} for the eddy viscosity $ν_t$ and the turbulent Prandtl number $Pr_t$, as both terms are divided by the mean velocity gradient according to the standard definition, in vertical buoyant flow. To predict the quantities of interest, e.g. the Nusselt number, a machine learning framework via symbolic regression is used with various cost functions, e.g. the mean velocity gradient, with the aid of the latest direct numerical simulation (DNS) dataset for vertical natural and mixed convection. The study has yielded two key outcomes: $(i)$ the new machine learnt algebraic models, as the reciprocal of $Pr_t$, successfully handle the infinity issue for both vertical natural and mixed convection; and $(ii)$ the proposed models with embedded coordinate frame invariance can be conveniently implemented in the Reynolds-averaged scalar equation and are proven to be robust and accurate in the current parameter space, where the Rayleigh number spans from $10^5$ to $10^9 $ for vertical natural convection and the bulk Richardson number $Ri_b $ is in the range of $ 0$ and $ 0.1$ for vertical mixed convection.

physics.flu-dyn↗

arXiv2021

Optimal sensor and actuator placement for feedback control of vortex shedding

We consider linear feedback control of the two-dimensional flow past a cylinder at low Reynolds numbers, with a particular focus on the optimal placement of a single sensor and a single actuator. To accommodate the high dimensionality of the flow we compute its leading resolvent forcing and response modes to enable the design of H2-optimal estimators and controllers. We then investigate three control problems: i) optimal estimation (OE) in which we measure the flow at a single location and estimate the entire flow; ii) full-state information control (FIC) in which we measure the entire flow but actuate at only one location; and iii) the overall feedback control problem in which a single sensor is available for measurement and a single actuator is available for control. We characterize the performance of these control arrangements over a range of sensor and actuator placements and discuss implications for effective feedback control when using a single sensor and a single actuator. The optimal sensor and actuator placements found for the OE and FIC problems are also compared to those found for the overall feedback control problem over a range of Reynolds numbers. This comparison reveals the key factors and conflicting trade-offs that limit feedback control performance.

physics.flu-dyn↗

arXiv2020

RANS Turbulence Model Development using CFD-Driven Machine Learning

This paper presents a novel CFD-driven machine learning framework to develop Reynolds-averaged Navier-Stokes (RANS) models. The CFD-driven training is an extension of the gene expression programming method (Weatheritt and Sandberg, 2016), but crucially the fitness of candidate models is now evaluated by running RANS calculations in an integrated way, rather than using an algebraic function. Unlike other data-driven methods that fit the Reynolds stresses of trained models to high-fidelity data, the cost function for the CFD-driven training can be defined based on any flow feature from the CFD results. This extends the applicability of the method especially when the training data is limited. Furthermore, the resulting model, which is the one providing the most accurate CFD results at the end of the training, inherently shows good performance in RANS calculations. To demonstrate the potential of this new method, the CFD-driven machine learning approach is applied to model development for wake mixing in turbomachines. A new model is trained based on a high-pressure turbine case and then tested for three additional cases, all representative of modern turbine nozzles. Despite the geometric configurations and operating conditions being different among the cases, the predicted wake mixing profiles are significantly improved in all of these a posteriori tests. Moreover, the model equation is explicitly given and available for analysis, thus it could be deduced that the enhanced wake prediction is predominantly due to the extra diffusion introduced by the CFD-driven model.

physics.flu-dyn↗

arXiv2019

Feedback control of vortex shedding using a resolvent-based modelling approach

An investigation of optimal feedback controllers' performance and robustness is carried out for vortex shedding behind a 2D cylinder at low Reynolds numbers. To facilitate controller design, we present an efficient modelling approach in which we utilise the resolvent operator to recast the linearised Navier-Stokes equations into an input-output form from which frequency responses can be computed. The difficulty of applying modern control design techniques to complex, high-dimensional flow systems is thus overcome by using low-order models identified from these frequency responses. The low-order models are used to design optimal control laws using $\mathcal{H}_{\infty}$ loop shaping. Two distinct control arrangements are considered, both of which employ a single-input and a single-output. In the first control arrangement, a velocity sensor located in the wake drives a pair of body forces near the cylinder. Complete suppression of shedding is observed up to a Reynolds number of $Re=110$. Due to the convective nature of vortex shedding and the corresponding time delays, we observe a fundamental trade-off: the sensor should be close enough to the cylinder to avoid any excessive time lag, but it should be kept sufficiently far from the cylinder to measure any unstable modes developing downstream. It is found that these two conflicting requirements become more difficult to satisfy for larger Reynolds numbers. In the second control arrangement, we consider a practical setup with a body-mounted force sensor and an actuator that oscillates the cylinder according to the lift measurement. It is shown that the system is stabilised only up to $Re=100$, and we demonstrate why the performance of the resulting feedback controllers deteriorates much more rapidly with increasing Reynolds number. The challenges of designing robust controllers for each control setup are also analysed and discussed.

physics.flu-dyn↗

arXiv2017

Compressible Invariant Solutions In Open Cavity Flows

A family of compressible exact periodic solutions is reported for the first time in an open cavity flow setup. These are found using a novel framework which permits the computation of such solutions in an arbitrary complex geometry. The periodic orbits arise from a synchronised concatenation of convective and acoustic events which strongly depend on the Mach number. This flow-acoustic interaction furnishes the periodic solutions with a remarkable stability and it is found to completely dominate the system's dynamics and the sound directivity. The periodic orbits, which could be called `exact Rossiter modes', collapse with a family of equilibrium solutions at a subcritical Hopf bifurcation, occurring in the quasi-incompressible regime. This shows compressibility has a destabilising effect in cavity flows, which we analyse in detail. By establishing a connection with previous 2D and 3D stability studies of cavity flows, we are able to isolate the effect of purely compressible two-dimensional flow phenomena across Mach number. A linear stability analysis of the equilibria provides insight into the compressible flow mechanisms responsible for the instability. A close look at the adjoint modes suggests that an eigenvalue merge occurs at a Mach number between 0.35 and 0.4, which boosts the receptivity of the leading mode and determines the onset of the unstable character of the system. The effect of the choice of base flow over the transition dynamics is also discussed, where in the present case, the frequencies associated to the leading eigenmodes show a strong connection with the frequencies of the periodic orbits at the same Mach numbers.

physics.flu-dyn↗

arXiv2016

Adjoint-based Optimal Flow Control for Compressible DNS

A novel adjoint-based framework oriented to optimal flow control in compressible direct numerical simulations is presented. Also, a new formulation of the adjoint characteristic boundary conditions is introduced, which enhances the stability of the adjoint simulations. The flow configuration chosen as a case study consists of a two dimensional open cavity flow with aspect ratio $L/H=3$ and Reynolds number $Re=5000$. This flow configuration is of particular interest, as the turbulent and chaotic nature of separated flows pushes the adjoint approach to its limit. The target of the flow actuation, defined as cost, is the reduction of the pressure fluctuations at the sensor location. To exploit the advantages of the adjoint method, a large number of control parameters is used. The control consists of an actuating sub-domain where a two-dimensional body force is applied at every point within the sub-volume. This results in a total of $2.256 \cdot 10^6$ control parameters. The final actuation achieved a successful reduction of the cost of $79.6\%$, by altering the directivity of the sound radiated by the trailing edge of the cavity and breaking up the large shear layer instabilities into smaller structures.

physics.comp-ph↗